What is (\partial)/(\partial x)((e^{-x/2})(4x^2)) ?

The answer to (\partial)/(\partial x)((e^{-x/2})(4x^2)) is 4(-(e^{-x/2}x^2)/2+2e^{-x/2}x)

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(\partial)/(\partial x)((e^{-x/2})(4x^2))

\frac{\partial}{\partial x} ((e^{- \frac{x}{2}}) (4 x^{2}))

4 (- \frac{e ^{- \frac{x}{2}} x ^{2}}{2} + 2 e^{- \frac{x}{2}} x)

Solution steps

\frac{\partial}{\partial x} ((e^{- \frac{x}{2}}) (4 x^{2}))

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 4 \frac{\partial}{\partial x} (e^{- \frac{x}{2}} x^{2})

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = e^{- \frac{x}{2}}, g = x^{2}

= 4 (\frac{\partial}{\partial x} (e^{- \frac{x}{2}}) x^{2} + \frac{\partial}{\partial x} (x^{2}) e^{- \frac{x}{2}})

\frac{\partial}{\partial x} (e^{- \frac{x}{2}}) = - \frac{1}{2} e^{- \frac{x}{2}}

\frac{\partial}{\partial x} (x^{2}) = 2 x

= 4 ((- \frac{1}{2} e^{- \frac{x}{2}}) x^{2} + 2 x e^{- \frac{x}{2}})

Simplify

4 ((- \frac{1}{2} e^{- \frac{x}{2}}) x^{2} + 2 x e^{- \frac{x}{2}}) : 4 (- \frac{e ^{- \frac{x}{2}} x ^{2}}{2} + 2 e^{- \frac{x}{2}} x)

= 4 (- \frac{e ^{- \frac{x}{2}} x ^{2}}{2} + 2 e^{- \frac{x}{2}} x)

f (x) = 2 x - 5 x^{\frac{3}{4}}

\int_{3}^{\infty} \frac{6}{x ^{2} - x} d x

\frac{\partial}{\partial y} (x^{2} - x y^{2} + 4 y^{5})

\frac{\partial}{\partial y} (sin (x) cos (y))

\frac{d}{d x} (\frac{3}{x ^{2} + 2})

What is (\partial)/(\partial x)((e^{-x/2})(4x^2)) ?
The answer to (\partial)/(\partial x)((e^{-x/2})(4x^2)) is 4(-(e^{-x/2}x^2)/2+2e^{-x/2}x)